Unfortunately it does not seem to be very precise.
When heating up at the beginning, the hotend overshoots by more than 10°C.
When the part cooling fan kicks in after the first layer it usually drops more than 10°C below the set temperature. I even had a security shut down due to a "heater failure" which was probably only because of this, because the heater works just fine.
During the rest of the print it fluctuates between +/- 5°C off the set temperature.

The bed PID seems to be okay though (+/- 0.3°C).

Is there a way to have the autotuning run more cycles to improve repeatability?

Well, I saved the settings using M500 and removed the existing M307 from my config.g but didn't load config-override.g with M501. To be honest, I thought config-override.g would be loaded automatically without calling M501.

Thank you for making this clear. I appended M501 to my config.g and will investigate if it works now.